Estimation of Ideal Points using Cast Vote Records

Introduction

How representative is a government of its people?

Cast Vote Records

Distribution of Cast Vote Records

Methodology

Continuous, latent outcome with categorical indicators –> IRT model

\[ \pi_{j,k(c)} = logit^{-1} (\gamma_{k(c)} (\alpha_j - \beta_{k(c)})) \]

where \(j\) indexes individuals and \(k(c)\) indexes a specific candidate \(c\) in a race \(k\).

• \(\gamma\) is the “discrimination” parameter

  • \(=1\) in the “Rasch” model, implies all races inform \(\alpha\) equally

• \(\alpha\) is the latent trait

• \(\beta\) is the “difficulty” parameter

• \(\kappa = \frac{\alpha}{\beta}\) is the cutpoint between each choice

Identification and Priors

• The scale and location of \(\alpha\) are unidentified (can add or multiply a constant with no change in likelihood)

  • Set prior to \(\mathcal{N}(0, 1)\)

• \(\gamma\)’s sign can also flip around with relation to the sign of \(\alpha\).

  • Constrain \(\gamma > 0\)

• Priors

  • Uninformative, standard priors for other parameters

  • Future work: Informative priors using party or NOMINATE/DIME scores to help with identification and speed convergence

Stan Code

// Priors
mu_beta ~ cauchy(0, 5);
alpha ~ std_normal();

for (k in 1:K){
  beta_raw[k,] ~ normal(0, sigma_beta);
  gamma_raw[k,] ~ lognormal(0, sigma_gamma);
}

sigma_beta ~ cauchy(0, 5);
sigma_gamma ~ cauchy(0, 5);

// Likelihood
for (j in 1:J) {
  matrix[K, C] logits = rep_matrix(-1e8, K, C);
  for (k in 1:K) {
    if (eligibility[j, k] == 1) {
      for (c in 1:C) {
        if (candidates[k, c] == 1) {
          logits[k, c] = gamma[k, c] * alpha[j] - (beta[k, c] + mu_beta);
        }
      }
      target += categorical_logit_lpmf(votes[j, k] | logits[k]');
    }
  }
}

Convergence

\(\pi_{i, j(c)}\) as a Binary Variable – Rasch

Binary Results - \(\alpha\)

Binary Results - \(\beta\) and \(\gamma\)

\(\pi_{i, j(c)}\) as a Categorical Variable – Rasch

Categorical Rasch Results

Categorical 2PL Results

Application to Representation

Work with the binary 2PL results, since they are easiest to interpret

References